MATLIS INJECTIVE MODULES

Abstract. In this paper, Matlis injective modules are introduced andstudied. It is shown that every R-module has a (special) Matlis injec-tive preenvelope over any ring R and every right R-module has a Matlisinjective envelope when R is a right Noetherian ring. Moreover, it isshown that every right R-module has an F ⊥ 1 -envelope when Ris a rightNoetherian ring and F is a class of injective right R-modules. 1. IntroductionThroughout this paper, R will denote an associative ring with identity andall modules will be unitary right R-modules.The motivation of this paper is from [4], where the notion of Whiteheadmodules was studied. Recall that an R-module M is called a Whitehead moduleor W-module if Ext 1R (M,R) = 0. We introduce the notion of Matlis injectivemodules as a dual notion of Whitehead modules in some sense. An R-moduleM is called Matlis injective if Ext 1R (E(R),M) = 0, where E(R) denotes theinjective envelope of R. Let R be an integral domain and Q its field of quotients,an R-module C is called Matlis cotorsion or weakly cotorson if Ext